Problem: Simplify and expand the following expression: $ \dfrac{y}{2y + 7}+\dfrac{4y + 2}{y + 5} $
Solution: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(2y + 7)(y + 5)$ Multiply the first term by $\dfrac{y + 5}{y + 5}$ $ \begin{align*} \dfrac{y}{2y + 7} \times \dfrac{y + 5}{y + 5} & = \dfrac{(y)(y + 5)}{(2y + 7)(y + 5)} \\ & = \dfrac{y^2 + 5y}{(2y + 7)(y + 5)}\end{align*} $ Multiply the second term by $\dfrac{2y + 7}{2y + 7}$ $ \begin{align*} \dfrac{4y + 2}{y + 5} \times \dfrac{2y + 7}{2y + 7} & = \dfrac{(4y + 2)(2y + 7)}{(y + 5)(2y + 7)} \\ & = \dfrac{8y^2 + 32y + 14}{(y + 5)(2y + 7)}\end{align*} $ Now we have: $ = \dfrac{y^2 + 5y}{(2y + 7)(y + 5)} + \dfrac{8y^2 + 32y + 14}{(y + 5)(2y + 7)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{y^2 + 5y + 8y^2 + 32y + 14}{(2y + 7)(y + 5)} $ $ = \dfrac{9y^2 + 37y + 14}{(2y + 7)(y + 5)}$ Expand the denominator: $ = \dfrac{9y^2 + 37y + 14}{2y^2 + 17y + 35}$